ADDENDUM2. The charge associated to any of these currents is related with the electrical flux at spatial infinity. Indeed one has:
$$Q = \int_{t=t_0} J^0 d^3x = \int_{t=t_0} \sum_{i=1}^3 F^{0i}\partial_i \Lambda d^3x = \int_{t=t_0} \partial_i\sum_{i=1}^3 F^{0i} \Lambda d^3x -
\int_{t=t_0} ( \sum_{i=1}^3 \partial_i F^{0i}) \Lambda d^3x \:.$$
As $\sum_{i=1}^3 \partial_i F^{0i} = -\partial_\mu F^{\mu 0}=0$, the last integral does not give any contribution and we have
$$Q = \int_{t=t_0} \partial_i\left(\Lambda \sum_{i=1}^3 F^{0i} \right) d^3x = \lim_{R\to +\infty}\oint_{t=t_0, |\vec{x}| =R} \Lambda \vec{E} \cdot \vec{n} \: dS\:.$$
If $\Lambda$ becomes constant in space outside a bounded region $\Omega_0$ and if, for instance, that constant does not vanish, $Q$ is just the flux of $\vec{E}$ at infinity up to a constant factor. In this case $Q$ is the electric charge up to a constant factor (as stressed by ramanujan_dirac in a comment below). In that case, however, $Q=0$ since we are dealing with the free EM field.

Thank you very much. Do you mean then that Noether's (first) theorem is applicable here, although it's not a global symmetry?
–
user46348May 12 '14 at 18:30

I provided a proof, what should I do further? ;)
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Valter MorettiMay 12 '14 at 18:32

I know... It's just that it conflicts with what others said here. Then can we say that as long as the symmetry is continuous, Noether's theorem applies? Also, why don't I see this current written anywhere?
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user46348May 12 '14 at 18:36

@user46348 I believe, but please correct me if I'm wrong, that Noether's first theorem in general does not apply to gauge transformations, it only applies to global transformations. However, apparently Noether's first theorem happens to work for $\mathrm{U(1)}$ electromagnetic theory.
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HunterMay 12 '14 at 19:02

@V.Moretti: So is the Noether's charge, the usual electric charge, and what you are stating is the conservation of electric charge?
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user7757May 13 '14 at 9:42

I asked myself the same question a while back, and this is what I came up with:

I assumed that the Lagrangian of the Maxwell action is of the form:
$$\mathcal{L} = \mathcal{L}(A_\mu,\,\partial_\nu A_\mu)$$
Then I assumed that under variations of the type $A_\mu \rightarrow A_\mu - \frac{1}{e}\partial_\mu\alpha$ where $\alpha\equiv \alpha(x)$ a local gauge parameter, the Lagrangian remains invariant:
\begin{align}
\delta\mathcal{L} &= \frac{\partial \mathcal{L}}{\partial A_\mu}\delta A_\mu + \frac{\partial \mathcal{L}}{\partial(\partial_\nu A_\mu)}\delta(\partial_\nu A_\mu)\\
&= \frac{\partial \mathcal{L}}{\partial A_\mu}({\textstyle \frac{-1}{e}\partial_\mu \alpha})+ \frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}({\textstyle \frac{-1}{e}\partial_\mu \partial_\mu \alpha})=0\,.
\end{align}
The in the first term, I use the equation of motion to replace $\frac{\partial\mathcal{L}}{\partial A_\mu}=\partial_\nu (\frac{\partial\mathcal{L}}{\partial (\partial_\nu A_\mu)})$ to get
\begin{equation}
\phantom{wwwwww}\delta\mathcal{L}=\partial_\nu\big(\frac{\partial\mathcal{L}}{\partial (\partial_\nu A_\mu)}\big)({\textstyle \frac{-1}{e}\partial_\mu \alpha})+\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}({\textstyle \frac{-1}{e}\partial_\mu \partial_\mu \alpha})=0\,.
\end{equation}
Now, since $\alpha(x)$ is an arbitrary smooth function, first derivative is independent of second derivative. So the two terms vanish separately.

Vanishing second term implies the tensor $\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}$ is antisymmetric (as it contracts with symmetric $\partial_\mu\partial_\nu \alpha$). So just define it $\frac{\partial \mathcal{L}}{\partial (\partial_\nu A_\mu)}:=F^{\nu\mu}$

Therefore, I conclude that Noether's theorem for gauge transformations is not a conservation law, but an equation of motion. And, the requirement of gauge symmetry imposes that $F^{\mu\nu}$ is antisymmetric.